What is the multiplicative inverse (reciprocal) of the integer 7?

Introduction / Context:
This question focuses on the idea of a multiplicative inverse, also known as a reciprocal, for a specific integer. Understanding inverses is essential in algebra because they are used to solve equations, simplify expressions and understand the structure of number systems. Here the integer in question is 7, and we must identify which option represents its multiplicative inverse.

Given Data / Assumptions:

• The integer given is 7, which is nonzero.
• The multiplicative inverse of a nonzero number a is the number that satisfies a * (its inverse) = 1.
• We are working in the real numbers.
• Division by zero is not allowed, so zero cannot be a reciprocal of any number.
• The goal is to find a number x such that 7 * x = 1.

Concept / Approach:
By definition, the multiplicative inverse of a number a is 1 / a. This definition directly gives the inverse once we know a. Therefore the multiplicative inverse of 7 is 1 / 7. Another way to think about it is to ask which number x satisfies 7 * x = 1. Solving for x gives x = 1 / 7, which matches the reciprocal definition. Any option that does not satisfy this equation or involves zero is incorrect.

Step-by-Step Solution:
Step 1: Let a be 7. The multiplicative inverse of a is defined as 1 / a.Step 2: Substitute a = 7 into this definition to get the candidate inverse 1 / 7.Step 3: Check the defining property of a multiplicative inverse: a * (1 / a) must equal 1 for nonzero a.Step 4: Compute 7 * (1 / 7). This is equal to 7 / 7, which simplifies to 1.Step 5: Therefore 1 / 7 satisfies the requirement and is the multiplicative inverse of 7.Step 6: Compare with other options to ensure they do not satisfy the same property.

Verification / Alternative check:
We can quickly test each option by multiplying it by 7. Option a is 1, and 7 * 1 = 7, not 1, so this fails. Option b is -1 / 7, and 7 * (-1 / 7) = -1, which is the negative of 1, so this also fails. Option c is 0, and 7 * 0 = 0, so it clearly fails and zero can never be a multiplicative inverse. Option d is 1 / 7, and as already shown, 7 * (1 / 7) = 1, so this passes the test. This confirms that 1 / 7 is the correct reciprocal.

Why Other Options Are Wrong:
The number 1 is the multiplicative identity, not the inverse of 7. It leaves 7 unchanged when multiplied, giving 7 instead of 1. The number -1 / 7 produces a product of -1, which fails the requirement. Zero is never a multiplicative inverse because any number multiplied by zero is zero, not 1. These observations rule out all the incorrect options.

Common Pitfalls:
Students sometimes confuse additive inverses and multiplicative inverses. The additive inverse of 7 is -7, since 7 + (-7) = 0, but that is unrelated to the multiplicative inverse, which focuses on products equal to 1. Another pitfall is thinking that the inverse of a whole number must also be a whole number, which is not true; many inverses are fractions.

Final Answer:
The multiplicative inverse of 7 is 1/7.